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Migration as an antidote to rent-seeking?Fabio Mariani

To cite this version:Fabio Mariani. Migration as an antidote to rent-seeking?. Journal of Development Economics, Else-vier, 2007, 84 (2), pp.609-630. �10.1016/j.jdeveco.2007.02.001�. �halshs-00186460�

Migration as an antidote to rent-seeking?∗

Fabio Mariani

UNIVERSITÉ PARIS 1 PANTHÉON-SORBONNE†

This draft: January 2007

Abstract

We develop a new mechanism through which skilled migration may influence eco-nomic performance in the sending country. If agents can choose between acting asrent-seekers and engaging in productive activities, and only productive skills are ex-portable, a positive probability of migration (to a more secure economy) reduces therelative expected returns from rent-seeking, thus decreasing the proportion of skilledworkers who opt for ”parasitic” activities. Such an improvement in the allocation oftalent may prevail over the loss of skilled workers due to outmigration. However, weshow that this result is not robust to the introduction of endogenous protection. If pro-ductive workers share their resources between accumulation of productive capital andinvestment in security, prospective migration may induce a weaker protection againstrent-seeking, which in turn might depress average income in the source economy.JEL classification: D72; F22; O1.Keywords: Rent-seeking; Skilled migration.

1 Introduction

Rent-seeking and skilled migration (the so-called ”brain drain”) are two common features

of less developed economies. Rent-seeking can take different forms: corruption, bribery,

excessive bureaucracy, crime, insufficient protection of property rights, lack of political cul-

ture and meritocracy, malfunctioning institutions, etc. As shown by Mauro (1995), who

supplies some data on the diffusion of corruption (and the like) around the World, develop-

ing countries are severely affected by this kind of pathology. On the other hand, Docquier

and Marfouk (2004) provide estimates of migration rates by skills and by sending country:

∗I am thankful to Raouf Boucekkine, Claude d’Aspremont, David de la Croix and Matthias Doepke fortheir precious comments on earlier drafts. The Co-Editor of this Journal, Gordon Hanson, and two anonymousreferees provided very helpful suggestions. I would also like to express my gratitude to seminar participantsat UCL (Louvain-la-Neuve), University of Auckland, FUNDP (Namur) and University of Evry for useful dis-cussion. All remaining errors are, of course, under my own responsibility. Financial support from the BelgianFrench-Speaking Community in the framework of the ARC Project "New Macroeconomic Perspectives on De-velopment" (Grant ARC 03/08-302), and from the Centre Cournot pour la Recherche en Economie ("RobertSolow" Scholarship) is very gratefully acknowledged.

†CES - Centre d’Economie de la Sorbonne; 106-112, bd. de l’Hôpital, F-75013 Paris (France). Ph.: +33 (0)144078350; fax: +33 (0)1 44078231. E-mail: fabio.mariani@univ-paris1.fr.

1

they observe that less developed countries display the highest brain drain rates and that,

in most cases, skilled migration has been increasing over the 1990-2000 decade.

Both of these problems are usually considered to be harmful for growth and economic

performance. Concerning rent-seeking, there is wide agreement that it exerts an adverse

effect on development; it can determine, for instance, a misallocation of productive abilities,

a wasteful employment of resources, or a decrease in investment rates (see Bardhan (1997)

for a survey, and Mauro (1995) for an empirical assessment). Skilled migration as well, since

the pioneering contribution of Bhagwati and Hamada (1974), has usually been regarded as

a threat to economic growth and considered as an impoverishing flight of human capital.

More recently, several models of brain drain have challenged this view, pointing out that a

positive, although limited, emigration rate may be beneficial for the source economy: the

opportunity to migrate would raise the relative expected returns to higher education, thus

determining an inducement effect on schooling decisions; this brain gain (in terms of higher

average human capital) may prevail over the actual brain drain. As examples of this new

approach, we can cite Mountford (1997), Stark and Wang (2002) and Beine et al. (2001).

However, until now theoretical models have studied rent-seeking and skilled migration

separately, without trying to establish a link between the two problems. Thus, the possible

consequences of their interaction for economic development have been neglected.

In the present paper we want to allow for such an interaction and see how migration

prospects may influence rent-seeking. To do that, we follow several models of rent-seeking

like Acemoglu (1995), Murphy et al. (1993), Torvik (2002), Mehlum et al. (2003), Baland and

François (2000), and study the endogenous career choice between carrying out productive

(entrepreneurial) activities and engaging in parasitic activities (rent-seeking). In this kind

of framework, we introduce a new mechanism: given that productive skills are typically

less country-specific and more exportable than rent-seeking abilities, a positive probability

of migration to a richer and more ”virtuous” country can decrease the relative expected

returns to rent-seeking and therefore reduce the attractiveness of this career option, thus

leading to a smaller proportion of rent-seekers in the source economy. We find that this

more favorable allocation of talent can more than compensate the loss of skilled workers

due to outmigration and a positive income-maximizing migration rate does exist.

We also check how the picture changes once we allow for endogenous protection: pro-

ductive agents may in fact employ some of their resources to impede predation, thus re-

ducing the fraction of income that rent-seekers can take away from them. In such a case,

probabilistic migration to a country with less rent-seeking may weaken the incentive to

raise defensive barriers, in favor of the accumulation of ”portable” skills1. Therefore, after

migration, rent-seeking may be less widespread but more effective, so that the net effect of

migration on income becomes ambiguous: we identify some conditions under which the

1Broadly speaking, prospective migration might induce ”weaker institutions” in the sending country: sincethey plan to work abroad, productive agents care less about the future quality of economic institutions at home.

2

result of beneficial migration can be reversed.

Our theoretical results are partly supported by existing data. We will show that, for

a cross-section of countries, it is possible to detect a positive (negative) correlation be-

tween brain drain rates and the proportion of people who get engaged in productive (rent-

seeking) careers, while the brain drain does not appear to be negatively correlated with the

after-migration total incidence of rent-seeking.

Let us also underline that our explanation for a beneficial brain drain is entirely new: in

fact, we look at a wasteful allocation of talent as a factor of underdevelopment, while the

existing literature on brain drain and growth has focused on the insufficient accumulation

of human capital. In both approaches there is room for a potentially beneficial effect of

migration, since the probability of migration is higher for the more productive group (en-

trepreneurs vs rent-seekers, skilled vs unskilled), whose endogenous size is thus positively

affected. However, the two underlying mechanisms are very different2.

Finally, the kind of analysis we are going to develop can be also related to a couple of

papers by Docquier and Rapoport (2003a, b) on migration and ethnic discrimination. They

study the effects of skilled migration on an economy populated by two groups - an ethnic

minority and a dominant ethnic group - of exogenous size, with the majority levying a

discriminative tax on the educated minority. However, our work looks at a different kind

of rent-extraction (unrelated to ethnic discrimination), treats the relative size of the groups

as endogenous (it being the result of a career choice, and not the deterministic consequence

of ethnic identity), and introduces the possibility of self-protection by the vexed group.

The paper is then organized as follows. After this Introduction, Section 2 is devoted

to the presentation of some stylized facts on the relation between skilled migration and

rent-seeking. Section 3 presents and solves the basic model. Endogenous protection and

its implications are analyzed in Section 4. Section 5 extends the model to allow for an

endogenous probability of migration. Finally, Section 6 concludes.

2 Stylized facts

Here, we want to use available cross-country data to check whether rent-seeking and the

probability of migration are correlated. Consistent with what has been put forward in the

Introduction, we look at two different aspects of rent-seeking: (i) the allocation of talent be-

tween rent-seeking and productive activities, that is the relative number of people engaged

in the two careers3, and (ii) the total incidence of rent-seeking, which takes also into account

the intensity of rent-seeking (i.e. the fraction of income that parasitic agents take away from

productive workers), and not only the size of the rent-seeking sector as measured by the

number of people who have opted for a rent-seeking career.

2In our model, the inducement effect determined by a positive chance of emigration acts through ”lessrent-seeking”, instead of ”higher education”.

3The expressions "allocation of talent" and "career choice" will be used interchangeably.

3

2.1 Skilled migration and career choices (allocation of talent)

As a proxy for the probability of (skilled) migration, we use the migration rate for people

with tertiary education or more, which we will sometimes refer to as the brain drain rate

throughout this Section. Data on migration rates by skills are provided by Docquier and

Marfouk (2004) for a sample of about 200 countries.

To account for the allocation of talent, we follow Murphy et al. (1991), who use college

enrollment in law as a measure of talent allocated to rent-seeking, and college enrollment

in engineering as a proxy for talent allocated to entrepreneurship. Notwithstanding some

evident limits4, an indicator of this type, focusing on educational choices, captures quite

well the responsiveness of agents to the relative returns of alternative career options. Al-

though the source is the same - enrollment in tertiary education by field (UNESCO, 1998,

1999, 2006), our measure of the allocation of talent between alternative careers is slightly

different from Murphy et al. (1991); more precisely, we use enrollment in the fields of sci-

ences and engineering5 as a proxy for the choice of a productive career, while to quantify

how many people opt for a rent-seeking career we consider enrollment in both law and reli-

gion6, since in several developing countries a sizable part of the bureaucracy has a religious

formation and quite often religious leaders act as rent-seekers.

We start by plotting tertiary education enrollment in the fields of science and engineer-

ing (in 19957) against brain drain rates (as of 1990) for the largest possible cross section of

countries8. The five-year lag has been chosen to deal with possible reverse causality issues:

a sizable brain drain may in fact be explained by an excess supply of productive skilled

workers. Moreover, in such a way most of the students are supposed to have taken the

decision about their field of specialization using also the information they had on current

migration possibilities. Figure 1 shows some positive correlation: countries with higher

skilled-migration rates are characterized by a higher fraction of students going for educa-

tion in "productive" fields. To summarize:

Stylized fact 1 Specialization in productive fields is positively correlated with skilled migration.

Figure 2 unveils the other side of the story: where the probability of migration is smaller,

there is a less favorable allocation of talent, and a larger fraction of people chooses to get

4Most lawyers and civil servants are supposed to be productive workers, and some bureaucracy is necessaryfor the efficient functioning of an economic system.

5Sciences are: maths, computer sciences, natural sciences and health-related sciences. Engineering andarchitecture are counted together.

6Religion is counted together with humanities and fine arts.7We have chosen 1995 (or the closest year, when data for 1995 are not available) since UNESCO data have

not the same degree of disaggregation in more recent years (2000 and 2004): for instance, it is not possible toknow exactly how many students are enrolled in law, since the new "broad field" of aggregation puts togetherlaw, business and social sciences.

8Subject to the following constraints: (i) the reliability rate of the brain drain estimates, as reported byDocquier and Marfouk (2004, p.27), must be higher than 80%, (ii) as suggested by Murphy et al. (1991, p. 523),only countries with more than 10000 students are included in the sample.

4

DZA

BEN

BFA

COG

CIV

EGY

ETH

MRT

MOR

NAM

NGA

ZAF

TZA

TGO

TUN

UGA

ZWECAN

CRI

CUB

DOM HND

MEX NIC

PAN

ARG CHL

COL

PRY

URY

CYP

HKG

IND

IRN

ISR

JAP

JOR

KWT

LBN

MMR

NPL

PAL

SAU

LKASYR

THA

TUR

ARE

YEM

ALB

AUTBEL

BGR

FRA

DEU

GRC

HUN

IRL

ITA

NLD

POLPRT

ESPCHE

GBRYUG

AUS

NZL

1020

3040

5060

enro

l. sc

i. &

eng

. (%

, 199

5)

0 10 20 30 40brain drain (%, 1990)

Figure 1: Brain drain and enrollment in productive fields for a cross-section of 68 countries

trained for rent-seeking jobs (law and religion, in our approximation). In other words:

Stylized fact 2 Specialization in rent-seeking fields is negatively correlated with skilled migration.

DZA

BFA

COG

CIVEGY

ETH

MRT

MOR

NAM

NGA

ZAF

TGO

TUN

UGA

ZWE

CANCRI

CUB

DOM HNDMEX

NICPAN

ARG

CHL

COLPRY URY

CYPHKG

IRN

JAP

JOR

KWT

LBNMMRNPL

PAL

SAULKA

SYR

THA

TUR

ARE

YEM

ALB

AUT

BELBGR

FRA

DEU

GRC

HUN

IRL

ITA

NLDPOL

PRT

ESP

CHEGBR

YUG

AUS

NZL

PNG

020

4060

enro

l. la

w &

rel

. (%

, 199

5)

0 10 20 30 40brain drain (%, 1990)

Figure 2: Brain drain and enrollment in rent-seeking fields for a cross-section of 65 countries

To summarize, we may say that available data seem to support the idea that a higher

migration probability corresponds to a better allocation of talent9.

2.2 Skilled migration and corruption ("perceived" rent-seeking)

In the Introduction, while talking about endogenous protection, we have put forward that

the total amount of rent-seeking can be higher after migration, even if the relative size of

9Before going on, let us underline that Figures 1 and 2 are built using all the available information, andnot only data relative to less developed countries. However, if we remove the most industrialized countriesfrom the sample, and keep only LDC’s, the quality of the results does not change: moreover, correlations areeven more evident. Results are also more clear-cut if we split the whole cross-section in regional subsamples(America, Africa, Asia-Pacific and Europe). All this additional evidence is available upon request.

5

the rent-seeking sector is reduced (i.e. even if a smaller proportion of workers has cho-

sen the "parasitic" career). But, how can we measure the total incidence of rent-seeking?

A suitable solution is represented by the Corruption Perception Index (CPI), published by

Transparency International10, which describes " ... how international businessmen and fi-

nancial journalists perceive corruption ..." in different countries around the World. As it is

built, this index takes into account both the size of the rent-seeking sector and the intensity

of rent-seeking activities (corruption). Since the CPI score measures transparency, on a 0-10

scale, we use (10 − CPI) as a proxy for rent-seeking.

Figure 3 shows that the correlation between skilled migration rates (in 2000) and per-

ceived rent-seeking (in 2005) is even, although weakly, positive. However, for brain drain

rates lower than 20 %, it is almost impossible to identify any kind of correlation. We are

then able to state the following:

DZA

AGO

BEN

BFA

BDICMRCOG

EGY

ETH KEN

MLIMOR

NAM

NGA

ZAF

SDN

TZAUGAZWE

CAN

CRICUB

DOM

SLV

HND

MEX

NIC

PAN

USA

ARGBOL

CHL

COL

ECUPRY

URY

AFGAZE

CYP

HKG

IND

IDN

IRN

ISR

JAP

JOR

KWT

LBN

MYS

MNG

MMR

NPLPAK

PAL PHL

SAULKA

SYRTHA

TUR

ARE

YEM

AUT

BLR

BEL

BGR

HRV

CZE

DNK

EST

FIN

FRA

DEU

GRC

HUN

IRL

ITA

LVA

MDA

NLDNOR

POL

PRT

RUS

SLK

SVN

ESP

SWE CHE

MKD

GBR

YUG

AUS

NZL

PNG

02

46

8re

nt−

seek

ing

(TI i

ndex

, 200

5)

0 10 20 30 40brain drain (%, 2000)

Figure 3: Brain drain and corruption (TI) for a cross-section of 95 countries

Stylized fact 3 There is no obvious correlation between corruption and skilled migration.

This confirms our guess that the allocation of talent to parasitic activities and the total inci-

dence of rent-seeking (corruption) may indeed follow different trajectories, thus justifying

an extension of our theoretical model to take into account endogenous protection.

Before going on, let us underline that a reverse causality is more than likely to play a role

in explaining a part of the relationships described in Figures 1-3. However, our theoretical

model only focuses on the effects of migration on rent-seeking.

3 The benchmark model

We start by considering a simple setting with exogenous protection: workers engaged in

productive activities cannot influence, by their choices, the fraction of income they lose

10Data are available on-line at http://www.transparency.org/policy_research/surveys_indices/cpi.

6

when they are subject to rent-seeking pressures.

3.1 Autarky

We introduce a simple model of rent-seeking à la Acemoglu (1995). The economy is pop-

ulated by homogeneous agents (workers). Each worker is endowed with a productive

capacity h, that we can assimilate to human capital, and is allowed to make a (career)

choice between engaging in productive activities and becoming a rent-seeker11. Then, if

she chooses to become a productive worker, she will be able to produce (gain) h unities of

income. However, part of this income may be lost due to the effect of rent-seeking aggres-

sion by parasitic agents. More precisely, we model the rent-seeker/producer interaction as

a matching process: every producer may deal with at most one rent-seeker, who will take

away an exogenous fraction q (with 0 < q < 1) of her product12; we will refer to q as the

"extortion rate" associated with rent-seeking. If an agent decides to act as a rent-seeker, and

if she actually comes in contact with a productive worker, she will earn a rent qh.

As a consequence of the matching process, we have that if rent-seekers exceed pro-

ducers in size, there will be someone among the former who will not find any productive

workers to predate. If the opposite is true, with rent-seekers being outnumbered by pro-

ducers, there will be some of the latter who will avoid extortion.

We identify by p and (1 − p) respectively the fraction of rent-seekers and the fraction of

productive workers that coexist in the economy. The equilibrium value of p arises endoge-

nously as a consequence of the career choice: it is determined as a value which equates

expected incomes from the two activities (agents are assumed to be risk-neutral). The ex-

pected revenue of the representative entrepreneur13 would be:

πE(p) =

{

(1 − qp

1−p )h if p < 12

(1 − q)h if p ≥ 12

. (1)

With p = 0, entrepreneurs would be able to keep all their production. Thereafter, an in-

crease in p would result in an increased probability p/(1 − p) to ”meet” a rent-seeker, and

then in a lower expected income. Finally, if p ≥ 1/2, predators are more numerous than

prey, and entrepreneurs keep only a fraction (1 − q) of their production, regardless of p.

From the point of view of parasitic agents, they are sure to meet a productive agent,

collecting qh, as long as p ≤ 1/2. For p > 1/2, a crowding-in effect (acting through a de-

creasing probability (1 − p)/p to be matched with a productive worker) will progressively

11In the real world, however, the same individual might carry out both rent-seeking and productive activi-ties: the fact that a bureaucrat collects bribes cannot exclude that she also performs some productive tasks.

12For instance, we may think of a corrupted bureaucrat asking for a bribe or simply of extortion carried outby an organized crime group.

13Throughout the rest of the paper, we will sometimes use the term ”entrepreneur” to label the representativeproductive worker, although she gains her income from her human capital, and not from physical capital.

7

erode parasitic rents down to 0 (for p tending to 1)14. Therefore, rent-seekers’ profits can be

expressed as:

πRS(p) =

{

qh if p < 12

q1−p

p h if p ≥ 12

. (2)

We assume that career choices are endogenous and irreversible, so that each worker has

to choose once and for all whether to engage in productive or parasitic activities.

If q < 1/2 there will not be any rent-seeking in the economy, since the entrepreneurial

career will turn out to be, for any p, more rewarding than the rent-seeking activity. But, as-

suming that q > 1/2, there will always be a positive fraction of rent-seekers in the economy,

and multiple equilibria may arise, as shown in Figure 4. Murphy et al. (1993), Acemoglu

(1995), Torvik (2002) and Mehlum et al. (2003), although resting on different explanations,

end up with the same kind of picture. And, as it has been widely discussed in those papers,

only one of the two equilibria with positive rent-seeking is stable (point E1 in Figure 4)15.

Our further analysis will focus on equilibria like E1; being characterized by a high fraction

of rent-seekers, they also better fit the case of less developed countries.

Figure 4: The allocation of talent: multiple equilibria

Solving πRS = πE, we can see that the high rent-seeking equilibrium we are dealing

with is characterized by p = q. An equilibrium like E2 would imply p = 1 − q.

14However, if we assume (somewhat more realistically) that each rent-seeker can deal with η (> 1) en-trepreneurs at once, the threshold value moves from 1/2 to 1/(1 + η).

15The stability analysis runs as follows (see for instance Murphy et al., 1993). Consider an equilibriumlike E2, and suppose now to be in a right neighborhood of E2: in this case returns to rent-seeking would behigher than returns from productive activities, more workers would rationally choose to become rent-seekers,p grows larger and the economy would be driven to E1. On the other side, assume that the economy is in a left

neighborhood of E2; in such a case expected income from entrepreneurship exceeds expected rents, workerswould have an incentive to get engaged in the productive career, and the economy would be driven towarda state with no rent-seeking at all (p = 0), which may also be considered as a stable equilibrium, technicallyspeaking. In a right neighborhood of E1, instead, profits would be larger than rents, justifying an incentive tochoose entrepreneurship: by consequence p would decrease, leading the economy back to E1.

8

3.2 Migration

To introduce migration, we assume that every agent who decides to work in the pro-

ductive sector has a positive chance of emigration m, while migration is not allowed for

rent-seekers. As we have already put forward in the Introduction, this assumption is not

unrealistic, since rent-seeking skills are very likely to be strongly country-specific, while

productive skills are not. In fact, high-skilled migration typically concerns scientists and

engineers, and not civil servants, lawyers and bureaucrats16.

Migration may be motivated by different reasons. Roughly speaking, the opportunity

to earn higher wages, net of migration costs and eventual skill depreciation, is the main rea-

son for labor mobility. However, many different factors may lie behind wage differentials.

Since we want to focus on the rent-seeking issue, we put aside technological explanations,

and assume that the foreign country is attractive because it protects property rights and/or

labor income more effectively than the sending country, so that qF < q. In Docquier and

Rapoport’s (2003) paper, wage differentials were neglected as well, to privilege an expla-

nation of migration based on the attractiveness of a discrimination-free country (from the

viewpoint of the ethnic minority).

The migration rate m, that for the moment we assume to be exogenous, can be deter-

mined by both internal and external policies. The emigration policy of the sending country

may in fact aim at controlling migration outflows, but it is more obvious to think that in-

ternational mobility is restricted by immigration authorities in the destination countries,

using instruments like quotas, visa policies, etc. Whatever the case, in the sending coun-

try productive workers face uncertainty in the sense that they have a probability m to be

accepted in the destination country, while with a probability (1 − m) they will be forced to

stay home17.

Once probabilistic migration is taken into account, and assuming for the sake of sim-

plicity that after migration the number of rent-seekers will continue to exceed the number

of entrepreneurs, the expected income of the representative entrepreneur would be given

by:

πE(p, m) = [(1 − m)(1 − q) + m(1 − qF)]h, (3)

while rent-seekers would expect to gain:

πRS(p, m) = q(1 − m)(1 − p)

ph. (4)

16A very nice example is provided by Murphy et al. (1991): Talleyrand was a bishop with a large tax incomein France, but when he escaped to the United States he acted as a successful entrepreneur. This is indeed a caseof ”career crossover” that we will not consider (career choices are irreversible in our setting), but illustratesquite well how even the brightest rent-seeking skills are not exportable. However, rent-seekers may be foreign-raised, as it still happens in many LDC’s.

17Being homogeneous, productive workers are randomly selected for migration. Otherwise, prospectivemigrants are likely to undergo some kind of selection by skills.

9

Probabilistic migration raises expected profits (through prospective foreign income)

and decreases rents (through tougher crowding-in: predators have a lower probability to

meet prey). As a consequence, we obtain the following Proposition:

Proposition 1 When the extortion rate q is exogenous, probabilistic migration to a more virtuous

country has a ”moralizing” effect on the allocation of talent in the home economy.

In fact, equating entrepreneurial profits (3) and rents (4), we get:

pM(m) =(1 − m)q

1 − mqF, (5)

which is lower than the proportion of rent-seekers without migration, p. However, only a

fraction (1 − m) of the (1 − pM) productive workers remains in their home country, and

the after-migration predator/prey ratio becomes:

γM(m) =q

1 − q + m(q − qF), (6)

which is always decreasing in m, and then it is lower than it would have been without

migration (γ = q/(1 − q)). Therefore, probabilistic migration to a country with less rent-

seeking induces a more favorable allocation of talent in the home economy.

This finding, suggesting that more migration implies less rent-seeking, echoes Docquier

and Rapoport’s (2003) results, according to which migration prospects have a ”protective”

effect on the minority. However, in their model this effect runs through a lower discrimina-

tion tax rate, while in ours predation is not weaker (q is unaffected), but only less diffuse.

3.3 An optimal migration rate?

Above, we have been able to isolate a beneficial impact linked to general migration. How-

ever, if we think that the rent-seeking/production career choice concerns specially high-

skilled workers, we could easily interpret it as a brain gain linked to high-skilled migration.

Different from the existing literature (see for instance Mountford, 1997, Beine et al., 2001,

or Stark and Wang, 2002), this positive effect is not related to the accumulation of human

capital through education. However, dealing with homogeneous agents, we have not con-

sidered the possible brain drain effect, i.e. the potential shrinking of the highly educated

group due to migration opportunity. To cope with this problem, we can build a very simple

extension of our basic model, introducing explicitly heterogeneity by skills.

More precisely, we assume that in our model economy there is also a class of low-skilled

individuals. These agents are not given the choice to become rent-seekers, since the extrac-

tion of rents is assumed to require skilled labor18. As workers, they are characterized by a

very low productivity that allows them to produce only at a subsistence level, so that their

18As in the case, for instance, of corrupted bureaucrats.

10

net income is equal to zero; as subsistence producers, they are ignored by rent-seekers19.

We assume that, before migration, for every skilled agent (rent-seeker or entrepreneur)

there are χ subsistence producers in the economy.

In such a stylized setting, we can look for an optimal (e)migration rate, i.e. that value

m∗ of m which maximizes a desirable objective function. We can propose two alternative

formulations for the objective function:

(i) the total number of productive skilled workers left in the home economy, that is given

by (1 − m)(1 − pM(m));

(ii) the after-migration average income of honest workers (productive skilled workers and

subsistence workers), i.e.(1 − m)[1 − pM(m)](1 − q)h

(1 − m)[1 − pM(m)] + χ.

Maximizing (1 − m)(1 − pM) could be desirable since growth, or prospective economic

performance in the sending economy may depend on the number of high-skilled, produc-

tive workers in the country20; it is also the simplest way to encompass both the brain drain

(through (1 − m)) and the brain gain (through (1 − pM)). Option (ii) looks like a very natu-

ral solution to identify an optimal m, it being very much in the spirit of a utilitarian welfare

function, where the utility (income) level of every group is weighted by its relative size.

It could also be interesting to consider a third kind of objective function: the after-

migration average income (including rents in the count). Such an objective would deserve

some attention as long as the identity of rent-seekers and entrepreneurs is anonymous a

priori, and the social planner feels committed to maximizing the well-being of the whole

population, regardless of their career choice: this case will be analyzed in Appendix B21.

However, we believe that functions (i) and (ii) are more interesting if we think that the

policy maker, on purely ethical grounds, should not care too much about the welfare of the

parasitic group; in this case the following Proposition holds:

Proposition 2 With probabilistic migration and an exogenous q, a strictly positive optimal migra-

tion rate does exist, if the intensity of rent-seeking in the foreign economy is sufficiently low.

In fact, under both objective functions (i) and (ii), m∗ is given by:

m∗ = max

0 ;1 − (1 − qF)

√

qq−qF

qF

, (7)

and it is easy to check that m∗ is strictly positive if qF < 2 − (1/q).

19In analogy with Docquier and Rapoport (2003): in their paper, uneducated minority members remain atthe subsistence level, thus escaping discriminatory taxation.

20Following for instance Mountford (1997), who considers an economy-wide growth externality which isassumed to depend on the relative number of educated workers.

21To some extent, it could also have been interesting to include the welfare of successful migrants in theobjective function. However, in conformity with the existing literature, we prefer to limit our analysis to thesending economy, to which emigrants do not belong any more.

11

It is interesting to notice that, if migration is allowed to a rent-free country (qF = 0),

there will always be a positive migration rate which maximizes income (or growth) in the

source economy. It is in fact given by m∗ = 1 − 1/2q.

Moreover, it is worth noting that m∗ is increasing in q: the higher the extortion rate in

the sending economy, the larger the optimal migration rate22.

4 Endogenous protection

The results we obtained in the previous Section look quite encouraging about a possibly

beneficial brain drain. Now, we want to check whether this kind of finding is robust to

the introduction of endogenous protection. Until now we have in fact assumed both q and

h to be exogenous, but this need not necessarily be the case. We could instead imagine

that there is a trade-off between the two. For example, in the case of physical capital, we

may say that the entrepreneur, before setting up her business, has to decide how to allocate

her resources between productive investments (machineries) and investments in protection

(like building walls to defend her property, hiring and training guards, or financing better

institutions). In the case of human capital, the scarce resource may be thought of as being

time, which can be allocated between accumulation of human capital (through education

or whatsoever) and some sort of political activity devoted to obtaining better protection of

property rights or to building a safer net of relationships (with bureaucrats, for instance).

Regardless of their nature, investments in protection are supposed to determine a lower q.

And in any case, a positive probability of migration to a more secure country will induce

entrepreneurs to invest more in productive skills and less in defensive activities, since they

obviously care less about security and the quality of institutions in their home country. As

a consequence, after migration the home economy may end up with a higher proportion

of productive workers (smaller size of the rent-seeking sector, p), but more intensity in

predation (higher q).

4.1 The extended model: autarky vs migration

To take explicitly into account the choice between these two different types of investments -

productive or defensive, we can give our model a simple two-period structure. The timing

of events goes as follows: the irreversible career choice is made at the beginning of the first

period (t), actual production and interactions between rent-seekers and entrepreneurs take

place during the second period (t + 1), while uncertainty about migration is cleared at the

beginning of the second period.

22In Appendix B we will also show that, not surprisingly, the optimal migration rate would be higher if rentswere included in the objective function. When adding rent-seekers to the computation of average income,there is no additional "drain effect" to be accounted for: parasitic agents do not migrate and average rents,depending directly on the predator/prey ratio, are always increasing in m

12

Therefore, the investment choice takes place before uncertainty about migration is cleared.

This is crucial for all the results that will be derived in this Section. An alternative timing

of events, with productive workers deciding about their protection effort after uncertainty

about migration is resolved, might be considered. However, it would not add anything

interesting to our previous analysis (carried out in Section 3 and based on an exogenous q),

since the investment in protection would not depend on the probability of migration23.

We also assume that during the first period every skilled worker who wants to engage

in productive activities is endowed with one unit of time: a fraction xt of this time can be

invested to build up future productive capacity according to ht+1 = xαt , where 0 < α ≤ 1;

the remainder of the time (1 − xt) can be used to reduce qt+1 through a defensive effort24.

Dropping time indexes, the function qt+1 = f (1 − xt) = F(xt) can be specified as:

q(x) = max[

q̄ − (1 − x)ρ;12

]

, (8)

where 1/2 < q̄ ≤ 1 and ρ ≥ 0. The parameter ρ accounts for the productivity of the

defensive effort (1 − x) in reducing the fraction of income that is lost due to rent-seeking,

as well as the opportunity cost of shifting resources from protection to accumulation of

productive skills; in this sense ρ may be related to the quality of institutions in the economy.

It is easy to check that: limx→0 q(x) = max[q̄ − ρ; 1/2] and limx→1 q(x) = q̄.

The function q(x) is depicted in Figure 5: the parameter ρ is crucial for determining its

slope. We would underline that for ρ = 0 the case of exogenous protection is reproduced:

q = q̄ does not depend on x. The lower bound 1/2 is binding as long as ρ > 1/2.

We assume for simplicity q̄ = 1. If no time is devoted to defense, the rent-seeker will

take all the production away from the worker; if all the time is spent building up defenses,

q may be driven down to 1/2 (entrepreneurs are not able to completely eliminate rent-

seeking relying only on their defensive effort). The representative worker will choose op-

timally x, in order to maximize her expected profits net of rent-seeking, that amount to

(1 − q(x))xα.

In autarky the optimal choice would be

x∗ =

{

α/(1 +α) if ρ ≤ 1/2

max[α/(1 +α); 1 − 1/2ρ] if ρ > 1/2. (9)

The possibility of corner solutions in the second case is due to the fact that it is never opti-

mal to select an x lower than 1 − 1/2ρ, since the implied decrease in production would not

determine any gain in protection (see Figure 5). To rule out this possibility (when ρ > 1/2),

23Under the assumption that rent-seekers continue to outnumber producers.24We do not model explicitly neither the choice of the aggressive effort made by the rent-seeker, nor her

accumulation of skills. Grossman and Kim (1995) have a closed-economy model with endogenous efforts onboth the defensive and the offensive side. Docquier and Rapoport (2003a, b) consider that the discriminativetax rate (that in their models plays the same role played by q in ours) is chosen by the rent-seeking ethnicmajority and cannot be influenced by the minority.

13

Figure 5: The function q(x)

we would simply need to impose an upper bound to the values that the parameter ρ may

assume, namely ρ ≤ (α + 1)/2.

If the possibility of migration is taken into account, the fraction of time devoted to human

capital accumulation becomes:

x∗M =

{

x⋄M if ρ ≤ 1/2

max[x⋄M; 1 − 1/2ρ] if ρ > 1/2, (10)

where

x⋄M = min[

α[m(1 − qF) + (1 − m)ρ]

(1 +α)(1 − m)ρ; 1

]

. (11)

Notice that x∗M = x⋄M as long as ρ ≤ ρ̄, where:

ρ̄ =(α + 1)− m(1 −α)− 2mαqF

2(1 − m),

while, to rule out corner solutions (x∗M = 1), we would simply need m to be lower than

m̄ = ρ/[α(1 − qF) + ρ]. Throughout the remainder of this Section, we will assume, for ease

of presentation, that both ρ ≤ ρ̄ and m < m̄ hold.

It is easy to see that x∗M is a decreasing function of qF, while it increases with m: more se-

curity abroad or a higher probability of migration, both these factors induce weaker invest-

ments in protection and more accumulation of productive skills25. In particular it is clear

25In our analysis, human capital can be accumulated only through private investments in education, andconsidering public education goes beyond our scopes. However, it is interesting to relate our contribution toPoutvaara (2004): in his model an increasing international applicability of a given type of education encouragesprivate investments in this kind of skills (similar to our model, where x increases with m), but " ... governmentsface an incentive to divert the provision of public education away from internationally applicable educationtoward country-specific skills. This would mean educating too few engineers, economists and doctors, and toomany lawyers. [...] It could even allow for a Pareto-improvement". However, in Poutvaara’s model, country-specific skills are not assumed to be employed in the rent-seeking sector.

14

that x∗M is larger than x∗ for every positive value of m: as we have already put forward,

easier migration puts more weight on expected income abroad, thus inducing a stronger

investment in portable skills and a smaller defensive effort.

Given the optimal (x, 1 − x) choice, we can find the equilibrium value of p, equating

expected profits and expected rents. In autarky, we solve:

(1 − x∗)ρ(x∗)α = [1 − (1 − x∗)ρ]1 − p

p(x∗)α , (12)

to get:

pǫ = 1 − ρ

(1 +α)=

α + (1 − ρ)

(1 +α). (13)

When migration is taken into account, the ”career-arbitrage” condition

[(1 − m)(1 − x∗M)ρ + m(1 − qF)] (x∗M)α = [1 − (1 − x∗M)ρ](1 − p)(1 − m)

p(x∗M)α (14)

determines:

pǫ,M(m) =α + (1 − ρ) 1−m

1−mqF

(1 +α). (15)

For every m ∈ (0, m̄), pǫ,M is lower than pǫ but, to assess the moralization effect of migra-

tion, we need to look at the after-migration predator/prey ratio:

γǫ,M(m) =pǫ,M

(1 − pǫ,M)(1 − m)=

α

(1 − m)+

(1 +α)(1 − ρ)

(1 − m)ρ + m(1 − qF). (16)

The following Proposition would then hold:

Proposition 3 Provided that ρ is low enough, the ”moralization” effect of probabilistic migration

is preserved also under the assumption of endogenous protection.

To see that, consider γǫ,M: it is decreasing in m, until m = m̌ where it gets a minimum. And

since:

m̌ =[(1 − ρ) +α][(1 − ρ)− qF]− (1 − qF)

√

α(1 +α)(1 − ρ)[(1 − ρ) − qF]

[(1 − ρ) +αqF][(1 − ρ)− qF], (17)

then: m̌ < m̄ for ρ > [√

α(1 +α)(1 − qF) + (αqF/2)2 −α(1 − qF/2)].

This result means that for low enough ρ’s, the predator/prey ratio is decreasing for

every admissible value of m, i.e. ∀m ∈ (0, m̄); non-monotonicity occurs only for higher

values of ρ. In the specific case of a rent-free destination country (qF = 0), we would

have that m̄ = ρ/(α + ρ) and γǫ,M(m) is monotonically decreasing in m as long as ρ ≤√

α(1 +α)−α. Let us just recall that ρ measures the opportunity cost of of diverting some

resources from defense to accumulation of productive capital; when both ρ and m are large,

property becomes poorly protected in the sending economy, since x is also high and this

has a sizable effect on q: this implies that the expected income of entrepreneurs is heavily

affected and the moralization effect stops working.

15

4.2 The optimal migration rate

Let us now see if a positive optimal (income-maximizing) emigration rate m∗ does exist,

when protection is assumed to be endogenous.

In straight analogy with the case of exogenous protection, and keeping in mind that x∗Mis in turn a function of m, the quantity to be maximized may be one of the following:

(i’) the after-migration total income of high-skilled producers (obviously, net of rent-seeking),

given by: ΨM(m) = (1 − m)(1 − pǫ,M(x∗M))(1 − q(x∗M))(x∗M)α;

(ii’) the after-migration average income of "honest" producers, i.e.:

(1 − m)[1 − pǫ,M(x∗M)](1 − q(x∗M))h(x∗M)

(1 − m)[1 − pǫ,M(x∗M)] + χ.

Let us consider the case with qF = 0: with an exogenous extortion rate q (see previous

Section), we were ensured that a strictly positive m∗ did exist. Under endogenous protec-

tion, things change; in fact, we can prove the following:

Proposition 4 If protection is endogenous and migration is directed to a rent-free economy, a

strictly positive value for m∗ may not exist if ρ exceeds a critical value ρ.

The claim of the above Proposition is easy to prove for function (i’). From ∂ΨM(m)/∂m = 0

(with ∂2ΨM(m)/∂m2 < 0), we obtain:

m∗ =2α(1 − ρ) + ρ(3 − 4ρ) +α2 −√

κ

4(1 − ρ)(α + ρ), (18)

where κ = α2(2 +α)2 − 2αρ[2(α2 − 1) +α]− ρ2[4α(1 +α)− 1].

It is easy to check that ∂m∗/∂ρ < 0: the optimal migration decreases with ρ. In particu-

lar, for m∗ to be strictly positive, we need simply that ρ < 1/2 = ρ. Therefore, for ρ ≥ 1/2,

m∗ would be equal to zero: if the sending economy is able to control its migration rate, the

optimal policy choice should be to keep its frontiers closed.

Furthermore, the condition ρ < 1/2 warrants the existence of a strictly positive value

of m (call it m̂), which maximizes the absolute number of skilled workers not engaged in

rent-seeking, given by (1 − m)(1 − pǫ,M(x∗M))26. Just notice that m∗ < m̂27, and ∂m̂/∂ρ < 0.

As it will be proved in Appendix A, the claim of Proposition 4 applies also to case (ii’),

while it cannot be extended to an objective function which includes rent-seeking incomes28

(see Appendix B).

26Maximizing this product is probably a more natural extension of (i) than (i’), but it would miss the pointthat the accumulation of productive resources is now endogenous, as well as the fraction of these resourcesthat is protected from rent-seeking.

27Both m∗ and m̂ are equal to zero, when ρ = 1/2.28In such a case a strictly positive m∗ does always exist. This result is not surprising: with endogenous

protection, productive workers produce more (h is higher) and set up weaker defenses (higher extortion rate q);as a consequence, a rent-seeker who comes in contact with an entrepreneur obtains a higher rent. In addition, aswas already the case with an exogenous q, rent-seekers also benefit from migration as long as the predator/preyratio decreases, there is less crowding-in and their chance to meet a productive worker increases.

16

This set of results (Propositions 3 and 4) conveys the message that, roughly speaking,

migration can better solve the rent-seeking problem, thereby improving economic perfor-

mance, only if the opportunity cost of diverting some resources (time) from protection to

accumulation of productive capital is not very high. If it is not the case (ρ ≥ 1/2), the

general result drawn from Section 3 is completely reversed: inducing weaker institutions

and less effective defense against rent-seeking, migration turns out to be harmful for eco-

nomic performance; the best policy option, for the developing country, would be to close

its frontiers to migration outflows.

5 Endogenous migration rate

Throughout the previous Section, we have assumed the probability of migration for pro-

ductive workers to be fixed at the level m, regardless of h and x. Indeed, it could be the

case that the chance of emigration is higher when the human capital of the prospective

migrant is higher. Some destination countries, for instance, tend to select immigrants by

skills: people with more years of education are more likely to be accepted.

To take this possibility into account, we can consider the probability of migration at time

t + 1 as being a function of the investment in productive skills made at time t; neglecting

time indexes, we would have m = z(x). For the sake of simplicity, we choose a linear

specification for z(x), such that m = σx, with 0 ≤ σ ≤ 1: the probability of migration

increases linearly with the quantity of time spent building up productive skills.

The parameter σ measures how efficiently education is transformed into a positive

chance to emigrate; in this framework, an increase in σ would correspond ceteris paribus

to an increased mobility of workers: for any given value of x, a higher σ translates into a

higher probability of migration. Just notice that σ = 0 corresponds to the autarkic case: the

probability of migration is zero, no matter how many years have been spent accumulating

human capital.

For the rest, the theoretical setup is the same as that of Section 4: in particular, we

assume that the extortion rate q depends positively on x (endogenous protection).

In this new framework, the first thing we may want to look at is the optimal quantity

of time devoted to human capital accumulation, x∗M′ . With qF = 0 (rent-free destination

country), it is given by:

x∗M′ = min

[

(1 +α)[ρ − (1 − ρ)σ ]−√

Λ

2(2 +α)ρσ; 1

]

, (19)

where Λ = (1 +α)2[ρ − (1 − ρ)σ ]2 − 4α(2 +α)ρ2σ .

Ignoring corner solutions, which can be ruled out as long as σ < ρ/(1 + α + ρ), we

can check that ∂x∗M′/∂σ > 0; when σ increases, productive workers will invest more in

education, since: (i) x becomes more decisive in determining their probability of migration,

and (ii) a higher probability of migration raises expected returns to education.

17

This result parallels the one obtained in Section 4, where the optimal x was a positive

function of m. Moreover we can see that, for σ = 0, the autarkic result with endogenous

protection is reproduced, in fact limσ→0 x∗M′ = α/(1 −α) (as in expression (9)).

Equating profits and rents, we obtain the equilibrium allocation of talent (the endoge-

nous size of the two groups) as a function of the parameter σ ; before migration takes place,

the proportion pǫ,M′(σ) of people engaging in a rent-seeking career is given by:

ρ{ρ[1 +α(σ − 1)2 − (6 −σ)σ ]− 2σ [σ − 3 +α(σ −α − 3)]− (1 +σ)√

Λ} +σ [σ(1 +α) +√

Λ]

2(2 +α)2ρσ.

(20)

Not surprisingly, we have that limσ→0 pǫ,M′(σ) = (α + 1 − ρ)/(1 +α) = pǫ (autarky).

But it is very important to notice that, in sharp contrast with our previous analysis, an

increased mobility of skilled workers does not reduce the fraction of educated agents in-

volved in rent-seeking activities. In fact, ∂pǫ,M′(σ)/∂σ is always positive29 (see Appendix

C for a proof) while, in the case of exogenous migration, we had ∂pM(m)/∂m < 0.

To understand how this result is determined, consider that when m is endogenous (m =

σx), an increase in σ will push productive workers to invest more in x (as stated above); a

larger x, however, will determine weaker protection, i.e. a higher extortion rate q. This lack

of protection will affect heavily the income of those productive agents who cannot migrate:

although the prospective income of successful migrants benefits from higher values of x

and σ , the expected income of productive workers (which is an average of the two) will not

raise that much. On the other hand, the expected income of rent-seekers shifts up notably,

whatever p, since the higher extortion rate q more than compensates a tougher crowding-

in (larger m). To sum up, both expected profits and expected rents increase; however, the

former shifts up by less, thus determining a lower p30.

Therefore, we have a first clue that the moralization effect of migration might not hold

any more, if the probability of migration is endogenous. Such an hypothesis is confirmed

by the inspection of the after-migration predator/prey ratio γǫ,M′(σ) = pǫ,M′(σ)/[(1 −pǫ,M′(σ))(1 − m(σ))], which is given by:

ρ2[1 +α − (3 +α)ρ] + 2ρ(ρ − 1)[1 +α + (2 +α)ρ]σ − (1 +α)(ρ − 1)3σ2 + [ρ(1 + ρ)− (ρ − 1)2σ ]√

Λ

2ρ[ρ2(σ − 1)2 +σ2 − 2ρσ(1 +σ)].

(21)

In fact, after checking that limσ→0 γǫ,M′(σ) = α +(1 +α)(1−ρ)/ρ = γǫ = limm→0 γǫ,M(m),

we can also see (Appendix C) that γǫ,M′(σ) is always increasing in σ . Therefore:

Proposition 5 When the probability of migration depends on skill accumulation, an increased mo-

bility of workers does not induce any moralization effect in the sending country.

29For the sake of completeness, consider that a corner solution (x∗M′ = 1) would determine pǫ,M′ (σ) = 1 −σ ,so that the result is reversed.

30Recall that the equilibrium value of p is found as the intersection of the two curves representing expectedprofits and expected rent-seeking income, respectively.

18

In analogy with our previous analysis, we would also be interested in assessing whether

an optimal migration rate exists. However, migration is now endogenous, and then we

may ask whether a strictly positive optimal value for σ (the exogenous parameter affecting

m) exists. For the sake of analytical tractability, we consider only the following objective

function:

ΞM′(σ) = (1 − m(x∗M′))(1 − pǫ,M′(x∗M′))(1 − q(x∗M′))(x∗M′)α ,

where x∗M′ is a function of σ , as we know from (19). The above expression represents the

after-migration total income of high-skilled producers, and corresponds to cases (i) and (i’)

in Sections 3 and 4.

It is immediately evident that the first three terms of the four that concur to determine

ΞM′(σ) are decreasing in σ . Therefore, it is unlikely that a strictly positive σ∗ may exist.

Although we cannot prove it analytically, in Appendix C we will use numerical simulations

to show that ΞM′(σ) is monotonically decreasing in σ for a wide range of parameter values.

Therefore, if the probability of migration depends on human capital, opening frontiers

unambiguously hurts the sending economy31.

We can now summarize our theoretical results. When the probability of migration does

not depend on skills, a strictly positive optimal migration rate m∗ always exists with exoge-

nous protection, while it exists only for some parameter configurations under endogenous

protection. Similarly, migration always induces a more favorable allocation of talent with

exogenous protection, and for values of m which are not too large in the case of endogenous

protection. However, if protection is endogenous and the likelihood of migration depends

on human capital accumulation, the moralization effect disappears and there is no strictly

positive optimal value for σ , i.e. for the parameter that determines the migration chance m.

6 Conclusions

In this paper we have explored the interplay between probabilistic migration and the en-

dogenous allocation of talent, as well as its consequences for economic performance in the

concerned country.

We started by presenting some stylized facts. Across countries, the skilled migration

rate appears to be positively (negatively) correlated with the fraction of educated agents

who specialize in productive (rent-seeking) fields. However, we see that this kind of corre-

lation is not preserved if instead of the relative size of parasitic and productive groups we

31It is important to underline that our representation of an endogenous probability of migration considersall the agents who have access to human capital enhancement as being the same vis-à-vis their educational pos-sibilities. It would be more realistic to introduce some heterogeneity, through a distribution of innate abilitiesin transforming x in h, and deal with the possible self-selection of migrants, but the model would be extremelydifficult to handle. Therefore, we propose to interpret our model as a polar case, useful to show how shiftingfrom random to endogenous migration affects the relationship between labor mobility and rent-seeking.

19

consider the total amount of rent-seeking, that we measure through a perceived corruption

index and which also involves an appreciation of rent-seeking intensity.

Then, we have built a simple model showing how, on purely theoretical grounds, prob-

abilistic migration can be expected to induce a moralizing effect in an economy affected

by a severe problem of rent-seeking and corruption: in such a case a positive income-

maximizing migration rate can be identified.

However, we have also pointed out that these quite optimistic findings rest on an ad-

mittedly simple assumption, namely that the effectiveness of rent-seeking is exogenously

determined. In fact, this kind of result may be weakened or may not hold any more once en-

dogenous protection is taken into account, depending on how efficiently productive agents

can manage to defend themselves against rent-seekers, and possibly on the quality of in-

stitutions and political culture in the developing country. Moreover, if not only protection

but also migration is made endogenous, there is even less room for a positive role of skilled

labor mobility in the perspective of reducing rent-seeking.

As interesting extensions, and possible directions for further research, we would sug-

gest: (i) to explore the growth implications of migration in a OLG model with joint rent-

seeking/income dynamics, (ii) to analyze possible strategic complementarities between bu-

reaucratic behavior and innovation, (iii) to endogenize the skill distribution of productive

workers, and (iv) to endogenize the offensive effort of rent-seekers (prospecting a decrease

in rents due to the lack of potential prey, migration may induce parasitic agents to soften

their pressure on productive workers).

References

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A Existence of a strictly positive m∗ with endogenous protection

(Proposition 4)

In Section 4, we have already shown under which conditions a strictly positive m∗ does ex-

ist, when the objective function is specified as in (i’). Let us now consider function (ii’):

in this case it is not possible to obtain an analytical expression for m∗. In fact, calling

21

ΦM(m) the after-migration average income of honest producers, we see that the equation

∂ΦM(m)/∂m = 0 does not admit closed form solutions. However, it is possible to identify

conditions under which a strictly positive m∗ may exist.

After recalling that in Section 4 we assumed m < ρ/[α(1 − qF) + ρ] to rule out corner

solutions, and that we are now working with qF = 0, it is easy to check that:

limm→0

∂ΦM(m)

∂m=

(

α

1 +α

)αρ(1 − 2ρ)χ

[ρ + χ(1 +α)]2

and

limm→ρ/(α+ρ)

∂ΦM(m)

∂m= − ρ(α + ρ)2

(1 +α)[αρ + (α + ρ)2χ].

It is clear that the second limit is always negative. If the first limit is positive, and that is

the case as long as ρ < 1/2, we are ensured that 0 < m∗ < ρ/(α + ρ). If ρ ≥ 1/2, a strictly

positive m∗ fails to exist.

B An alternative specification for the social objective function:

the after-migration average income (including rents)

The results we have obtained in Section 3 and 4 were based on the assumption that the

utility of rent-seekers should not be included in the social objective function. However,

this needs not necessarily be the case. Suppose in fact that the identity of rent-seekers

and entrepreneurs is anonymous a priori and thus the social planner feels committed to

maximizing the well-being of the whole population, regardless of their career choice. In

such a case an appropriate objective function would be the after-migration average income

(including rents in the count).

B.1 Exogenous protection

Assuming for the moment that the extortion rate q is exogenous, the objective function

writes as:

{(1 − m)[1 − pM(m)](1 − q) + pM(m)[(1 − m)(1 − pM(m))/pM(m)]q}h

(1 − m)[1 − pM(m)] + pM(m) + χ;

the optimal migration rate is then given by:

m∗′ = max

0 ;q + χ − (1 − qF)

√

qχ(1+χ)q−qF

q + χqF

, (22)

that is strictly positive provided that qF < [q2 + χ(2q − 1)]/[q(1 + χ)].

With qF = 0, we have that m∗′ = [q + χ −√

χ(1 + χ)]/q; this optimal m is always

positive, since we assumed q > 1/2.

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It is worth noting that m∗′ , like m∗ (as given by (7)), is increasing in q: the higher the ex-

tortion rate in the sending economy, the larger the optimal migration rate. Not surprisingly,

we can also check that m∗′ > m∗. This depends on the fact that, when adding rent-seekers

to the computation of average income, there is no additional "drain effect" to be accounted

for: parasitic agents do not migrate and average rents, depending directly on the preda-

tor/prey ratio (see the objective function above), are always increasing in m.

B.2 Endogenous protection

The suitable objective function, call it ΘM(m), now becomes:

{(1 − m)[1 − pǫ,M(x∗M)](1 − q(x∗M)) + pǫ,M(x∗M)[(1 − m)(1 − pǫ,M(x∗M))/pǫ,M(x∗M)]q(x∗M)}h(x∗M)

(1 − m)[1 − pǫ,M(x∗M)] + pǫ,M(x∗M) + χ.

In this case it is not possible to obtain an analytical expression for the optimal migration

rate m∗′ , since the equation ∂ΘM(m)/∂m = 0 does not admit closed form solutions.

However, it is possible to identify conditions under which a strictly positive m∗′ exists.

In fact, assuming that qF = 0, it can be shown that:

limm→0

∂ΘM(m)

∂m=

αα(1 +α)−α−2[(1 +α − ρ)2 + (1 +α)(1 +α − 2ρ)χ]

(1 + χ)2

and

limm→ρ/(α+ρ)

∂ΘM(m)

∂m=

(α + ρ)2[α(α +α2 − ρ2) + (1 +α)(α − ρ)(α + ρ)χ]

(1 +α)[α(α + 2ρ) + (α + ρ)2χ]2.

A sufficient condition for the first limit to be positive is that ρ < (1 +α)/2. But this coin-

cides with the restriction ρ < ρ̄ in Section 4. We are then ensured that m∗′ is strictly positive,

as long as corner solutions are ruled out. Moreover, if ρ > α√

(1 +α)(1 + χ)/[α + (1 +α)χ]

the second limit is negative, and we are sure that 0 < m∗ < ρ/(α + ρ).

Therefore, Proposition 4 does not hold any more if the objective function includes rents

as well. To understand why, consider that when protection is endogenous, productive

workers produce more (h is higher) and set up weaker defenses (higher extortion rate q); as

a consequence, a rent-seeker who comes in contact with an entrepreneur is able to obtain

a higher rent. Moreover, as was already the case with an exogenous q, rent-seekers also

benefit from migration as long as the predator/prey ratio decreases, there is less crowding-

in and the probability to meet a productive worker increases. From the viewpoint of a

policy maker who also cares about rent-seekers’ welfare, the optimal m would never be 0.

C Endogenous migration: proofs and numerical simulations

First, we want to prove that both pǫ,M′(σ) and γǫ,M′(σ) (as defined in Section 5) are increas-

ing in σ . For ease of exposition, we will present our computations in the case of α = 1.

However, our results fully generalize to other values of α between 0 and 1.

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We can write:

∂pǫ,M′(σ)

∂σ= − [ρ(σ − 1)−σ ]{2ρ2 + (ρ − 4)ρσ + 2(ρ − 1)2σ2 − 2[ρ + (ρ − 1)σ ]

√ζ}

18ρσ2√

ζ, (23)

where ζ = [ρ + (ρ − 1)σ ]2 − 3ρ2σ .

For derivative (23) to be positive, we need σ > ρ/(ρ− 1), which is always verified with

ρ < 1, since σ is assumed to take positive values.

A fortiori, also γǫ,M′(σ) is increasing in σ . In fact:

γǫ,M′(σ) =pǫ,M′(σ)

(1 − pǫ,M′(σ))(1 − m(σ)),

and we have already proved that the numerator pǫ,M′(σ) increases with σ , while the de-

nominator is a negative function of σ .

In addition, we present the output of some numerical simulations, in order to show that

a strictly positive value of σ which maximizes function ΞM′(σ) (as specified in Section 5)

seems not to exist.

After fixing ρ = 0.4, in Figure 6 we have drawn ΞM′(σ) for α = 0.2, α = 0.55 and

α = 0.9, respectively. In all cases, ΞM′(σ) is monotonically decreasing in σ . This holds true

for every 0 < α ≤ 1, and whatever ρ.

0.025 0.05 0.075 0.1 0.125 0.15 0.175sigma

0.02

0.04

0.06

Xi

Figure 6: The function ΞM′(σ)

Although we are not able to prove it analytically, it is then possible to say that a strictly

positive σ∗ should not exist. This stands in sharp contrast with the results we obtained

throughout Sections 3 and 4 (exogenous protection).

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